Greedy Forwarding in Dynamic Scale-Free Networks Embedded in Hyperbolic Metric Spaces

1
Greedy Forwarding inDynamic Scale-Free
NetworksEmbedded in Hyperbolic Metric Spaces

• Dmitri KrioukovCAIDA/UCSD
• Joint work with
• F. Papadopoulos, M. Boguñá, A. Vahdat

2
Outline

• Model of scale-free networks embeddedin
  hyperbolic metric spaces
• Greedy forwarding in the model
• Conclusion

Motivation
3
Motivation

• Routing overhead is a serious scaling limitation
  in many networks (Internet, wireless, overlay/P2P
  networks, etc.)
• Search for infinitely scalable routing without
  any overhead
• Do not propagate any information about changing
  topology
• Route without any global topology knowledge,
  using only local information
• How is it possible?

3
4
Greedy geometric forwarding as routing using only
local information

• Network topology is embedded in a geometric space
• To reach a destination, each node forwards the
  packet to the neighbor that is closest to the
  destination in the space

4
5
Hidden space visualized
6
Desired properties of greedy forwarding, and
related metrics

• Property 1 Greedy routes should never get stuck
  at local minima, nodes that do not have any
  neighbor closer to the destination than
  themselves
• Success ratio, percentage of successful greedy
  paths reaching their destinations, should be
  close to 1
• Property 2 Greedy paths should be close to
  shortest paths
• Stretch, ratio of the lengths of greedy to
  shortest paths, should be also close to 1
• Property 3 Even if topology changes, success
  ratio and stretch should stay close to 1 without
  any recomputation (e.g., without nodes changing
  their positions in the space)

6
7
Problem formulation (high-level)

• Find a combination of network topology and
  underlying geometric space which would satisfy
  these desired properties
• Any suggestions?
• Nature offers some many dynamic networks in
  nature and society do route information without
  any topology knowledge (brain, regulatory, social
  networks, etc.)
• All these complex networks have power-law degree
  distributions (scale-free) and strong clustering
  (many triangular subgraphs)
• Lets focus on these topologies (which, luckily,
  also characterize the Internet and P2P networks)
• But what about the underlying space?

7
8
Conjecture space is hyperbolic

• Nodes in real complex networks can often be
  classified hierarchically
• Hierarchies are tree-like structures
• Hyperbolic geometry is the geometry of tree-like
  structures
• Formally trees embed almost isometrically in
  hyperbolic spaces, not in Euclidean ones

8
9
Main hyperbolic property the exponential
expansion of space

• Circle length and disc area grow with their
  radius R as
  eR
• They are exactly
  2? sinh R 2? (cosh R ?
  1)
• The numbers of nodes in a tree at or within R
  hops from the root grow as
  bRwhere b is the tree branching
  factor
• The metric structures of hyperbolic spaces and
  trees are essentially the same

9
10
Problem formulation (low-level)

• Verify the conjecture check if hyperbolic
  geometry, in the simplest possible settings, can
  naturally give rise to scale-free, strongly
  clustered topologies
• Check if greedy forwarding satisfies the desired
  properties in the resulting embedding

10
11
Outline

• Motivation
• Greedy forwarding in the model
• Conclusion

Model of scale-free networks embeddedin
hyperbolic metric spaces
12
The model
13
The model (cont.)
14
14
15
Average node degree at distance r from the disc
center
16
Node degree distribution
17
Model vs. AS Internet
18
Growing networks

• The model can be adjusted for networks growing in
  hyperbolic spaces
• All results stay the same

18
19
Outline

• Motivation
• Model of scale-free networks embeddedin
  hyperbolic metric spaces
• Conclusion

Greedy forwarding in the model
19
20
Two greedy forwarding algorithms

• Original Greedy Forwarding (OGF) select closest
  neighbor to destination, drop the packet if no
  one closer than current hop
• Modified Greedy Forwarding (MGF) select closest
  neighbor to destination, drop the packet if a
  node sees it twice

21
Property 1success ratio
22
Property 2average and maximum stretch
23
Property 3 Robustness of greedy forwarding
w.r.t. network dynamics

• Scenario 1 Randomly remove a percentage of links
  and compute the new success ratio
• Scenario 2 Remove a link and compute the
  percentage of paths that were going through it
  and are still successful (that is, the percentage
  of paths that found a by-pass)

24
Percentage of successful paths (dynamic networks,
scenario 1)
25
Percentage of successful paths (dynamic networks,
scenario 2)
26
Shortest paths in scale-free graphs and
hyperbolic spaces
26
27
Outline

• Motivation
• Model of scale-free networks embeddedin
  hyperbolic metric spaces
• Greedy forwarding in the model

Conclusion
27
28
Conclusion (low-level)

• Hyperbolic geometry naturally explains the two
  main topological characteristics of complex
  networks
• scale-free degree distributions
• strong clustering
• Greedy forwarding in complex networks embedded in
  hyperbolic spaces is exceptionally efficient

28
29
Conclusion (mid-level)

• Complex network topologies are naturally
  congruent with hyperbolic geometries
• Greedy paths follow shortest paths that
  approximately follow geodesics in the hyperbolic
  space
• Both topology and geometry are tree-like
• This congruency is robust w.r.t. topology
  dynamics
• There are many link/node-disjoint shortest paths
  between the same source and destination that
  satisfy the above property
• Strong clustering (many by-passes) boosts up the
  path diversity
• If some of shortest paths are damaged by link
  failures, many others remain available, and
  greedy routing still finds them

29
30
Conclusion (high-level)

• To efficiently route without topology knowledge,
  the topology should be both hierarchical
  (tree-like) and have high path diversity (not
  like a tree)
• Complex networks do borrow the best out of these
  two seemingly mutually-exclusive worlds
• Hidden hyperbolic geometry naturally explains how
  this balance is achieved

30
31
Implications

• Greedy forwarding mechanisms in these settings
  may offer virtually infinitely scalable
  information dissemination (routing) strategies
  for communication networks
• Zero communication costs (no routing updates!)
• Constant routing table sizes (coordinates in the
  space)
• No stretch (all paths are shortest, stretch1)

31
32
Applications

• Internet routing (hard) need to reverse the
  problem and find an embedding for a given
  Internet topology first
• Overlay networks
• with underlay (easier examples existing P2P)
  have freedom of constructing a name space and its
  embedding according to the model, so that all the
  desired properties are satisfied
• without underlay (harder examples CCN, pocket
  switching) need to make sure that the underlay
  network topology and its dynamics are congruent
  with the overlay name space and its dynamics

32